MMSE estimation in Gaussian channels plays a central role in several information-theoretic applications (e.g., [1]}, [2]}, [3]}, [4]}, [5]}). The MMSE dimension [6]} is a measure of nonlinearity of the MMSE estimator. The first-order derivative of the conditional expectation in Gaussian channels has been treated in [7]}. In particular, formula (REF ) is generalized in [7]} to the multivariate case. To the best of our knowledge, no generalization such as (REF ) to the higher-order derivatives exists in the literature.
